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#### Original function
$Min\sum_{i=1}^{m} \sum_{d=1}^{7} \sum_{t=0}^{23} A_{i,d,t} \times CK_{d,t}$

#### Taking into account multiple users

$Min\sum_{u=1}^{n}\sum_{i=1}^{m} \sum_{d=1}^{7} \sum_{t=0}^{23} A_{u,i,d,t} \times CK_{d,t}$

* $u$ - user index
* $n$ - total number of users


#### Taking into system load

The assumption is that the energy distributor would not want extremely high number of appliances turned on at a certain time, while at other times the system is idle.

##### Total used appliances in a time-slot $t$ on a day $d$.

* $UA_{d,t} = \sum_{u=1}^{n}\sum_{i=1}^{m} A_{u,i,d,t}$ 

##### Mean appliances per day $d$

* $M_d = \frac{1}{24}\sum_{t=0}^{23} UA_{d,t}$


##### Load inbalance per day $d$

* $L_d = \sum_{t=0}^{23} |UA_{d,t}-M_d|$

##### Total load inbalance $d$

* $TL = \sum_{d=1}^{7} L_d$

Final function:

$Min[(\sum_{u=1}^{n}\sum_{i=1}^{m} \sum_{d=1}^{7} \sum_{t=0}^{23} A_{u,i,d,t} \times CK_{d,t})+cTL]$, where $c$ is some normalization coefficient (maybe experimentally determined)

